Circle Through 3 Points Calculator (Centre, Radius, Equation)
By Santaji GadepatilinUpdated November 10, 2026
Any three points that are not on one straight line lie on exactly one circle, the circumcircle of the triangle they form. This circle through 3 points calculator finds the centre (h, k) and radius r of that circle, writes its standard equation (x − h)² + (y − k)² = r² and its general form x² + y² + Dx + Ey + F = 0, and gives the circumference and area — handy for fitting an arc to three measured points.
How to use the circle through 3 points calculator
Enter the coordinates of three points that are not on a straight line. The calculator gives the centre (h, k), the radius r, the coefficients of the general equation x² + y² + Dx + Ey + F = 0, and the circumference and area of the circle. The determinant term d is zero when the points are collinear — then no circle exists.
Circle through three points formulas
d = 2[x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)] · sᵢ = xᵢ² + yᵢ²
h = [s₁(y₂ − y₃) + s₂(y₃ − y₁) + s₃(y₁ − y₂)] ÷ d · k = [s₁(x₃ − x₂) + s₂(x₁ − x₃) + s₃(x₂ − x₁)] ÷ d
r = √[(x₁ − h)² + (y₁ − k)²] · (x − h)² + (y − k)² = r²
Worked example
Points (1, 1), (2, 4) and (5, 3):
- s₁ = 2, s₂ = 20, s₃ = 34
- d = 2[1 × (4 − 3) + 2 × (3 − 1) + 5 × (1 − 4)] = −20
- h = [2 × 1 + 20 × 2 + 34 × (−3)] ÷ (−20) = 3, k = [2 × 3 + 20 × (−4) + 34 × 1] ÷ (−20) = 2
- r = √[(1 − 3)² + (1 − 2)²] = √5 = 2.236068
- Equation: (x − 3)² + (y − 2)² = 5, or x² + y² − 6x − 4y + 8 = 0
- Circumference = 14.049629, area = 15.707963
Example point sets
| Points | Centre (h, k) | Radius |
|---|---|---|
| (0, 0), (4, 0), (0, 3) | (2, 1.5) | 2.5 |
| (1, 0), (0, 1), (−1, 0) | (0, 0) | 1 |
| (0, 0), (2, 2), (4, 0) | (2, 0) | 2 |
| (−3, 4), (4, 3), (5, 0) | (0, 0) | 5 |
| (1, 1), (2, 4), (5, 3) | (3, 2) | 2.236 |
| (0, 0), (1, 1), (2, 2) | collinear — no circle | — |
For a right triangle the centre is the midpoint of the hypotenuse — the first row is the 3-4-5 triangle with radius 2.5.
Practical notes
- Measuring an arc or a pipe bend: pick the three points as far apart as possible along the arc; points close together make the radius very sensitive to measurement errors.
- The centre is the circumcentre of the triangle formed by the points — inside the triangle if it is acute, on the hypotenuse if it is right-angled and outside if it is obtuse.
- To check a result, the distance from the centre to each point should equal r — use the distance and midpoint calculator.
- For more than three noisy points, a least-squares circle fit gives a better estimate than any single set of three.
Frequently asked questions
How do you find the equation of a circle through three points?
The centre is equally far from all three points, so it lies where the perpendicular bisectors of two chords meet. Solving those two linear equations gives h = [s₁(y₂ − y₃) + s₂(y₃ − y₁) + s₃(y₁ − y₂)] ÷ d and k = [s₁(x₃ − x₂) + s₂(x₁ − x₃) + s₃(x₂ − x₁)] ÷ d, where sᵢ = xᵢ² + yᵢ² and d = 2[x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)]. The radius is the distance from the centre to any of the points.
What if the three points are on a straight line?
Then d = 0 and no circle passes through them — the perpendicular bisectors are parallel. The calculator shows an infinite or undefined centre and radius.
What is the general form of the equation of a circle?
x² + y² + Dx + Ey + F = 0, with D = −2h, E = −2k and F = h² + k² − r². You can also find D, E and F directly by substituting the three points and solving the three linear equations.
How do I find the radius of an arc from three measured points?
Measure the coordinates of three points on the arc — for example both ends and the middle — and enter them. For a chord of length c and height (sagitta) s the radius is also r = c² ÷ (8s) + s ÷ 2.